Abelian variety isogeny class 2.97.abi_sm over $\F_{97}$

• Label: 2.97.abi_sm
• Base field: $\F_{97}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{97}$
Dimension:	$2$
L-polynomial:	$1 - 34 x + 480 x^{2} - 3298 x^{3} + 9409 x^{4}$
Frobenius angles:	$\pm0.100083112680$, $\pm0.217693055951$
Angle rank:	$2$ (numerical)
Number field:	4.0.827712.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	12

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6558$	$86709876$	$832755622302$	$7838451049734096$	$73743945254977114158$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$64$	$9214$	$912436$	$88540774$	$8587518724$	$832973615854$	$80798293725808$	$7837433616007678$	$760231058584757200$	$73742412692078320654$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

• $y^2=58 x^6+69 x^5+14 x^4+36 x^3+95 x^2+71 x+91$
• $y^2=43 x^6+33 x^5+15 x^4+26 x^3+94 x^2+63 x+66$
• $y^2=77 x^6+50 x^5+11 x^4+51 x^3+12 x^2+30 x+41$
• $y^2=33 x^6+57 x^5+89 x^4+x^3+96 x^2+24 x+77$
• $y^2=85 x^6+61 x^5+25 x^4+78 x^3+77 x^2+13 x+39$
• $y^2=87 x^6+75 x^5+23 x^4+9 x^3+30 x^2+76 x$
• $y^2=63 x^6+62 x^5+37 x^4+66 x^3+64 x^2+22 x+59$
• $y^2=51 x^6+55 x^5+52 x^4+95 x^3+60 x^2+87 x+22$
• $y^2=48 x^6+59 x^5+95 x^4+74 x^3+64 x^2+15 x+92$
• $y^2=24 x^6+66 x^5+21 x^4+37 x^3+22 x^2+44 x+41$
• $y^2=79 x^6+51 x^5+36 x^4+33 x^3+34 x^2+6 x$
• $y^2=14 x^6+11 x^5+42 x^4+11 x^3+10 x^2+67 x+78$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{97}$.

Endomorphism algebra over $\F_{97}$

The endomorphism algebra of this simple isogeny class is 4.0.827712.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.97.bi_sm	$2$	(not in LMFDB)